High School

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It is possible for a vector space to contain exactly 2,479,220 vectors. (You must enter T or F -- True and False will not work.)

a) True
b) False

Answer :

The statement that a vector space can contain exactly 2,479,220 vectors is false; a non-trivial vector space contains infinitely many vectors because it is closed under vector addition and scalar multiplication.

The statement that a vector space can contain exactly 2,479,220 vectors is false. A vector space contains infinitely many vectors or none at all, depending on whether the space is non-trivial (has at least one vector).

By definition, a vector space is closed under vector addition and scalar multiplication, which means that by combining vectors through these operations, we can always generate an infinite number of additional vectors, as long as the space is not trivial (consisting of just the zero vector).

Furthermore, a vector space is n-dimensional if the maximum number of linearly independent vectors in the space is n. However, within an n-dimensional vector space, there are infinitely many vectors because each vector can be represented by a unique combination of these n linearly independent vectors scaled by real or complex numbers.

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